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Multicell multiuser massive MIMO channel estimation and MPSK signal block detection applying twodimensional compressed sensing
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 238 (2018)
Abstract
For the uplink multicell massive multipleinput multipleoutput (MIMO) block fading systems, a twodimensional smoothed l_{0} channel estimation method (2DSL0CE) with the aid of virtual channel representation is firstly exploited in this paper, which can jointly estimate the desired multiuser channels of the target cell and the interference links from neighbor cells without inducing pilot contamination. Then, a 2DSL0 signal detection method (2DSL0SD) with the aid of sparse decomposing and the modified 2D sl_{0} recovery algorithm is proposed, which can jointly decode Mary phaseshift keying (MPSK) signal block for whole desired users. Moreover, an improved 2DSL0SD is also proposed to remove multiuser interference of neighbor cells in high SNR scenario. Simulation results show that the 2DSL0CE method can remove performance floor induced by pilot contamination and need less pilot overhead than the conventional least square (LS) method. When detecting QPSK signal blocks at 12 dB SNR, the 2DSL0SD method with perfect channel state information (CSI) can obtain 10^{−2} BER. Moreover, in the case of 8PSK signals, the 2DSL0SD joining with the 2DSL0CE can obtain 10^{−2} BER at 20 dB SNR.
Introduction
The high energy and spectrum efficiency of massive multipleinput multipleoutput (MIMO) systems heavily build on the premise that the base stations (BS) obtain channel state information (CSI) with reasonable quality, which is generally estimated via pilot sequences [1]. However, in the uplink massive MIMO systems, the pilot overhead demanded should be proportional to the number of users and would be prohibitively large as the number of users increase. In the uplink multicell massive MIMO, this results in pilot contamination as the same pilot sequences have to be reused by neighbor cells to serve a large number of users [2]. Moreover, the pilot contamination is a major limiting factor to system performance [3]. Hence, the massive MIMO urgently needs efficient channel estimation scheme without producing pilot contamination and requiring too much pilot overhead. Based on the estimated CSI, the signals received at base stations are typically detected through linear methods with low complexity, such as zeroforcing [4–6] and matched filter [7, 8]. However, the performances of linear detector are typically far inferior to the optimal maximum likelihood (ML) detector whose computational complexity exponentially scales up with the signal constellation size and the number of antennas [9]. Thus, the development of computationally efficient and reliable detector for massive MIMO also needs to be thoroughly addressed [10].
In the past few years, several types of schemes have been exploited to mitigate or reduce the impact of pilot contamination in multicell massive MIMO systems. (1) Semiblind or blind approaches, such as [11–14]—the eigenvalue decompositionbased method with a short training sequence was proposed in [11]. Hu et al. [12] proposed a semiblind method without requiring the statistical information of channels. Another lowcomplexity semiblind approach was proposed in [13], which the received signal are firstly projected onto the subspace with minimal interference, then alternatively refined the channel estimation and detected the data symbols. Applying the theory of large random matrices, [14] proposed a blind pilot decontamination with subspace projection. (2) Optimization design of nonorthogonal pilot signals, such as [15–18]—when training slots are not large enough to construct the orthogonal pilot signals, [15] exploits a pilot design criterion and shows that the line packing on a complex Grassmannian manifold is the optimization scheme, which is based on minimal mean square error (MMSE) estimator. A generalized Welchbound equalitybased pilot signal design method is proposed in [16], which has low correlation coefficients and ensures the network to satisfy the requirement of user capacity. For a given pilot length, [17] proposes an alternating minimizationbased pilot design algorithm. (3) The precodingbased approaches, such as [19–21]—a MMSEbased precoding is exploited in [19] to alleviate the impact of pilot contamination. A pilot contamination mitigation method along with zeroforcing precoding is proposed in [20], which can generate orthogonal pilot signals across neighboring cells through multiplying the ZadoffChu sequences elementwise with a specific orthogonal variable spreading factor code.
Some significant efforts have been made to reduce the pilot overhead for massive MIMO systems, which can be divided into two broad categories. (1) Lowrank channel covariance matrices based methods, such as [22–24]—the finite scattering environment and small angular spread result in high correlation of different paths between the user and the BS [25–29] and lowrank channel covariance matrix. Through exploiting the correlation characteristic of channel vectors, the joint spatial division and multiplexing (JSDM) was proposed in [23] which significantly reduced the overhead of downlink training and uplink feedback for frequency division duplexing (FDD) massive MIMO systems. When the number of pilot signals is no less than the rank of channel covariance matrix and the noise interference disappear, [24] proves that the MMSE estimator can recovery channel vectors exactly. (2) Compressed channel sensing method—exploiting the channel sparsity and applying the compressed sensing (CS) to reduce the overhead of CSI feedback has been investigated in [30–32]. A spare channel estimation method applying Gaussianmixture Bayesian learning has been proposed in [33] to estimate the whole channel parameters including the desired and interference links, which can mitigate pilot contamination and reduce pilot overhead, but every time, the approach just can estimate the channel response at one beam.
An iterative MIMO detector with relaxed ML constraints using sparse decomposition has been proposed to preserve a low computational cost even increase the signal size, but the method just suit to detect a vector [34]. In block fading systems, the detection target at the BS usually is a multiuser data frame, i.e., a twodimensional (2D) signal block. To detect the 2D signals, the method in [34] should run the decoding process many times or convert the 2D signal detection problem to a vector detection problem. However, the converting method will substantially increase the required memory and processing load which would make it become noncompetitive when applied to massive MIMO block fading systems. For example, the converting approach can be represented as:
with A∈R^{60×200}, B∈R^{30×100}, x=vec(X), and y=vec(Y), which results in Φ=B⊗A with dimensions 1800×20,000. The signs of vec() and \(\bigotimes \) denote vectorization of a matrix and Kronecker product, respectively.
In this paper, the multiuser channel estimation problem and the multiuser signal decoding problem in uplink massive MIMO systems are modelled as twodimensional sparse signal recovery problems in compressed sensing, respectively. Although [35, 36] researched the 2D compressed sensing channel estimation schemes for massive MIMO, it is the FDD model discussed in [35, 36] which is different from the time division duplexing (TDD) multicell multiuser system model considered in this paper. The main contributions of this paper are summarized as follows:

We propose a channel estimation method named as 2DSL0CE applying the twodimensional smoothed l_{0}norm compressed sensing recovery algorithm [37, 38], which are able to jointly estimate multiuser CSI. Using virtual channel representation, the 2DSL0CE formulates the joint channel estimation problem, comprising both the target and interference channels, as a 2D sparse signal reconstruction problem in CS, which not only can mitigate the pilot contamination but also can significantly reduce pilot overhead.

We propose a signal detection method named as 2DSL0SD using our modified 2DSL0 algorithm, which can decode multiuser Mary phaseshift keying (MPSK) signal block. Applying sparse decomposition, the 2DSL0SD models the detection problem of multiuser signal block as a 2D sparse signal reconstruction problem whose elements are binaries {0,1}. Moreover, in the high SNR scenario, through exploiting the estimated CSI of interference links, an improved 2DSL0SD is also proposed to remove the decoding bottleneck induced by interference from neighboring cells.
The remaining paper is organized as follows. The system model and the least square (LS) channel estimation methods are described in Section 2. Section 3 models the multicell multiuser channel estimation problem as a twodimensional compressed sensing problem and describes the proposed channel estimation algorithm. Section 4 models the signal decoding problem of multiuser as a 2D sparse signal recovery problem and presents the steps of the proposed 2DSL0SD method. Section 5 gives and analyzes the numerical results. Section 6 draws a conclusion of the whole paper.
Notations: diag(x) represents a diagonal matrix with diagonal elements being the vector x. Superscripts T and † denote the transpose and pseudoinverse, respectively. CN(0,1) denotes complex Gaussian variables with zero mean and unit variance. \(\bigotimes \) denotes Kronecker product. Vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively.
System model
Consider a multicell massive MIMO system in which each target cell shares the same frequency band with L−1 adjacent cells. Each cell has one BS with a uniform linear array (ULA) of M antennas that serves K (K<<M) single antenna users. The uplink channel vector from the kth terminal in ith cell to jth BS is modelled as:
where g_{jik} is the fast fading vector, and largescale fading factor β_{jik} describes the quasistatic shadow fading and the path loss. Different channel vectors are assumed to be independent. Consequently, the channel matrix between the K users in ith cell and the jth BS can be represented as:
with H_{ji}=[h_{ji1},⋯,h_{jiK}], G_{ji}=[g_{ji1},⋯,g_{jiK}] and \(\mathbf {D}_{ji}=\text {diag}\left (\sqrt {\beta _{ji1}},\cdots,\sqrt {\beta _{jiK}}\right)\).
In a block fading channel, the training signal received at the jth BS becomes:
where P_{tr} denotes the training signaltonoise ratio (SNR), the rows of \(\mathbf {X}_{i}^{tr}\in C^{K\times T_{tr}}\) are the pilot sequences of ith cell, and \(\mathbf {N}_{j}^{tr}\) is the noise with i.i.d. elements distributed as CN(0,1).
The jth cell is assumed as the target cell. One natural choice to find a channel estimate based on the training signal without employing any prior information is the LS method which is given by:
where the second term denotes pilot contamination resulting in the same orthogonal pilots reused by adjacent cells.
2DSL0CE channel estimation method
The key idea of our exploited channel estimation is to explore the sparsity in the virtual channel representation, which applies spatial beams at fixed virtual directions to characterize the physical channel matrix. The virtual channel matrix \(\mathbf {G}_{ji}^{v}\) can be linked to the above described physical channel matrix G_{ji} by the following transformation:
where A_{R}=[a_{R}(θ_{1}),⋯,a_{R}(θ_{M})] with the receiver response vectors given by:
The direction θ_{m} is related to the physical angle ϕ_{m}∈[−π/2,π/2] as θ_{m}=dsin(ϕ_{m})/λ with λ being the carrier wavelength and d being the antenna spacing [39]. We uniformly sample the principal θ period to set the virtual spatial angles, i.e., θ_{m}=m/M, and resulting in an M×M unitary discrete Fourier transform matrix A_{R}. The element \(\mathbf {G}_{ji,mk}^{v}\) of M×K matrix \(\mathbf {G}_{ji}^{v}\) represents the coupling gain from the kth terminal to the mth virtual receive angle θ_{m}. Therefore, the element will be zero when there is no corresponding coupling, and the \(\mathbf {G}_{ji}^{v}\) will be a sparse matrix when the number of nonzero elements is much smaller than that of the total elements.
Substituting (6) into (4) yields the following received training signal at the jth BS:
Furthermore, taking the transpose operation to (8), we can obtain:
Now, based on the linear model Y=XGA_{R}+N, the channel estimation problem is modelled as a 2D sparse signal reconstruction problem in compressed sensing. Then, we estimate H_{j}=[H_{j1},⋯,H_{jL}] based on Y, X, and A_{R}, using the 2DSL0 sparse reconstruction algorithm [38]. The proposed channel estimation method 2DSL0CE is summarized in Algorithm 1. Different from other types of compressed sensing recovery algorithms, the SL_{0} and 2D−SL_{0} applied the following function to approximate the l_{0}norm of b, i.e., b_{0}.
where b∈R^{M×1} is a sparse vector, and the parameter σ determines the quality of the approximation and how smooth the function F_{σ}(b). Consequently, the minimum l_{0}norm solution can be found by maximizing F_{σ}(b). In Algorithm 1, steps 2–9 gradually decrease the value of σ and maximize the objective function for each value of σ.
2DSL0SD signal detection method
In a block fading scenario, the received data signal at the jth BS can be written as:
where \(\mathbf {Y}_{j}\in \mathcal {C}^{M\times N}\) is the received data, \(\sqrt {P_{\text {data}}}\) denotes the uplink SNR, X_{i} denotes the transmitted data matrix of the ith cell whose element is selected from a finite alphabet constellation defined as {s_{1},⋯,s_{Q}} with Q being the finite alphabet cardinal, N_{j} is the noise with elements distributed as CN(01), and W_{j} represents the noise plus interference faced by the received data of jth BS.
The transmitted symbol of the ith cell can be sparse represented as (12) through exploiting the prior knowledge that each transmitted element X_{i,mn} belongs to a discrete and finite alphabet constellation:
where X_{i,mn} denotes the nth symbol of mth user in ith cell, s=[s_{1},⋯,s_{Q}] is the discrete and finite constellation vector, and e_{i,mn}=[e_{i,mn}(s_{1}),⋯,e_{i,mn}(s_{Q})]^{T} with e_{i,mn}(s_{q}) being equal to 1 if X_{i,mn}=s_{q} or 0 otherwise (1≤q≤Q). Applying this sparse representation into all symbols, the transmitted data matrix in the ith cell can be expressed in function of a sparse matrix as:
where \(\mathbf {B}_{s}=\mathbf {I}_{K} \bigotimes \mathbf {s}\) is a block diagonal matrix of size K×KQ, and the nth column of the matrix E_{i} is [(e_{i,1n})^{T},⋯,(e_{i,Kn})^{T}]^{T}.
Substituting (13) into (11) generates
with \(\mathbf {A}_{j}=\sqrt {P_{\text {data}}}\mathbf {H}_{jj}\mathbf {B}_{s}\). The detection problem of signal block has been modelled as a 2D sparse binary {0,1} reconstruction problem in CS, then based on Y_{j}, A_{j}, and I_{N}, the signal block X_{j} can be detected using the modify 2DSL0 algorithm which suits to reconstruct 2D sparse binary {0,1} signal. The process of 2DSL0SD is summarized in Algorithm 2. Because the elements of E_{j} needed to be recovered are 0 or 1, but the elements recovered by the original 2DSL0 algorithm are not exactly 0 or 1, step 10 of Algorithm 2 is added to find which element of \(\hat {\mathbf {e}}_{j,mn}\) maybe 1 with highest probability and reset such element to 1 and others to 0.
In the high SNR scenario, such as N_{j}→0, it can be observed from (11) that the main factor restricting the decoding performance is not the noise but the interference from neighboring cells. Thus, the performance of 2DSL0SD will meet floor as the SNR increases. In order to resolve this problem, the CSI of interference links are exploited and an improved 2DSL0SD method is proposed, which does not treat the signals from neighbor cells as interference, moreover, decodes them jointly with the desired signals. Specifically, exploiting the CSI of whole links estimated by the above proposed 2DSL0CE method, the received data signal at the jth BS in (11) can be rewritten as:
Moreover, substituting (13) into (15) can generate:
Now, the decoding problem without multiuser interference has also been modelled as a 2D sparse {0,1} signal reconstruction problem, which has the same form as that of (14), and can be solved through the processes of Algorithm 2 whose A_{j} needs to be replaced by A. Hereafter, the improved 2DSL0SD with interference cancel is named as 2DSL0SDIC. Comparing (16) with (14), it can be observed that the recovery object of the 2DSL0SDIC is E, including not only the E_{j} of the objective cell but also the E_{i} of the L−1 neighbor cells (i=1,⋯L,i≠j), and the noise is the only interference source.
It is worth noting that the proposed 2DSL0SD is only suitable to decode constant modulus signal, such as MPSK. Since there is only one element of e_{i,mn} in (12) equaling to 1 and the others are zeros, Algorithm 2 needs a ruler to reset the values of the estimated \(\hat {\mathbf {e}}_{i,mn}\). In step 10 of Algorithm 2, the ruler is that the element of \(\hat {\mathbf {e}}_{j,mn}\) with the largest real part is viewed as such element whose value is 1 with the highest probability. Such ruler requires that the elements of s in (12) have the same modulus.
Numerical results and discussion
The spectral efficiency and estimation accuracy of the proposed 2DSL0CE sparse channel estimation method and the decode performance of the proposed 2DSL0SD are investigated. A multicell scenario with L cells sharing the same frequency band is considered. The fading coefficient is modelled as β_{jik}=z_{jik}/(r_{jik}/r_{h})^{α}, in which \(z_{jik}\thicksim ln N\left (0,\sigma _{\text {shadow}}^{2}\right)\) is a lognormal random variable, r_{jik} denotes the distance between the BS and the corresponding terminal, and r_{h} is the cellhole radius. The number of nonzero elements in each column of \(G_{ji}^{v}\), which means the number of multipath, is assumed to S, whose positions are randomly selected and values are generated through CN(0,1). The pilot sequence of each user is randomly generated using the complex normal distribution and then is normalized to unity. The normalized mean square error (NMSE) defined as \(\frac {1}{N_{\text {MC}}}{\sum \nolimits }_{i=1}^{N_{\text {MC}}}\frac {\\hat {H_{i}}H_{i}\_{F}^{2}}{\H_{i}\_{F}^{2}}\) is used to evaluate the estimation accuracy, where N_{MC} means the number of MonteCarlo simulations, H_{i} and \(\hat {H_{i}}\) are the actual and estimated channel impulse response of ith MonteCarlo trial, respectively. The parameters of the system and the 2DSL0 algorithm are summarized in Table 1.
Firstly, at the scenario of S=5, the NMSE of the 2DSL0CE and the bit error rate (BER) of the 2DSL0SD detection QPSK signals with various CSI, including perfect and estimated CSI, are investigated. The whole number of multicell users is 20×7=140; thus, the LS method should require orthogonal pilot sequences with length being not less than 140 to avoid pilot contamination. Figure 1 shows that the 2DSL0CE method applying random pilot sequences with a length of 12 can outperform LS with 140 pilots nearly 5 dB. Moreover, the NMSE of 2DSL0CE is approximate linear reducing with the SNR increasing when the pilot length is not less than 12. While in the case that 20 orthogonal pilot sequences are reused by the objective cell and neighbor cells, Fig. 1 also shows that the LS method will meet a performance floor caused by pilot contamination. In a block fading scenario with signal length N=200, Fig. 2 plots the BER performance of the 2DSL0SD detection QPSK signal block using various CSIs. Applying perfect CSI at 10 dB SNR, the 2DSL0SD can approach near to 10^{−2} BER. Moreover, its BER is approximate linear reducing with the SNR increasing, which shows its reliable detection ability. Using the CSI estimated through the 2DSL0CE, where each user applies random nonorthogonal pilot sequence with length being 32, the obtained BER of 2DSL0SD at 15 dB SNR is slightly better than 10^{−2}, which shows joint the 2DSL0CE with 2DSL0SD for channel estimation and decoding QPSK signals is a feasible scheme.
Then, applying perfect CSI and estimated CSI through 2DSL0CE, respectively, Figs. 3 and 4 show the BER of 2DSL0SD and its improved version 2DSL0SDIC detecting various PSK data block with length N=200. In both cases, the BER of 2DSL0SD is better than 2DSL0SDIC within a SNR threshold value. This is because the received energy of interference data is usually smaller than that of target data, which induces more difficult to decode interference data. The decoding object in 2DSL0SDIC includes the target and interference data, which means the decoding result of two parts will affect each other. Thus, in the lowSNR scenario, the low correct decoding probability of interference data leads to 2DSL0SDIC has higher BER than 2DSL0SD. With the SNR increasing and exceeding a threshold value, the 2DSL0SD will gradually meet a performance bound induced by interference. However, the BER of 2DSL0SDIC will be still approximately linear reducing owing to gradually high correct decoding probability of interference data. In theory, with the distance of the two neighbor signal points in constellation diagram gradually reducing, it will be more difficult to decode the signals correctly. In Figs. 3 and 4, it can be observed that the BERs of QPSK, 8PSK, and 16PSK are gradually increasing at the same SNR.
Finally, the NMSE of 2DSL0CE method with fixed pilot length at various scenarios, including various sparsity and number of users in each cell, is studied in Figs. 5 and 6, respectively. In the case of each cell having 20 users, applying random pilot sequences with fixed length of 26, Fig. 5 shows that the NMSE is approximately linear reducing with the SNR increasing when the S is not larger than 20. In the case of S=5 and pilot length being 26, Fig. 6 shows that the NMSE is approximately linear reducing with the SNR increasing when the number of users in each cell is not larger than 50. In order to recovery a 1D Esparse Flength vector successfully with high probability, [40] presents that the number of measurements needed is of order \(\mathcal {O}(E\text {log}(F/E))\). In the 2D cases, from the best performance plots in Figs. 5 and 6, it can be obtained that the number of measurements are 2.15 and 2.23 times of KLSlog(M/S), respectively, where the value of KLS is the number of nonzero elements in 2D signals. Thus, ensuring the original 2D signal can be recovered with high probability, the number of measurements needed is also of order \(\mathcal {O}(E\text {log}(F/E))\), where the E and F denote the numbers of nonzero elements and whole elements in the 2D signal, respectively. In practical, the above ruler can be used to guide for setting pilot length.
Conclusions
This paper has investigated the two challenging problems for block fading massive MIMO systems. The one is to exploit efficient uplink channel estimation method that requires acceptable pilot overhead and can mitigate pilot contamination. The other one is to jointly decode multiuser data block. The joint multiuser channel estimation and data block detection problems have both been modelled and solved as a 2D sparse signal reconstruction problem in the CS framework. More specifically, through using a linear virtual channel representation for ULA, the 2DSL0CE compressed channel sensing method is proposed, which needs less pilot overhead than LS method, and can jointly estimate the desired and interference uplinks. Through sparse representation in a finite alphabet set for each transmitted data symbol, the 2DSL0SD data decoding method is proposed which can simultaneously decode a MPSK data block for multiuser. Simulation results demonstrate that joint the 2DSL0CE with 2DSL0SD to estimate channel and decode MPSK data for multiuser massive MIMO is a feasible scheme.
Abbreviations
 2DSL0:

Twodimensional smoothed l_{0}
 2DSL0CE:

Twodimensional smoothed l_{0}norm channel estimation
 2DSL0SD:

Twodimensional smoothed l_{0}norm signal detection
 2DSL0SDIC:

Twodimensional smoothed l_{0}norm signal detection with interference cancel
 BER:

Bit error rate
 BS:

Base station
 CS:

Compressed sensing
 CSI:

Channel state information
 FDD:

Frequency division duplexing
 JSDM:

Joint spatial division and multiplexing
 LS:

Least square
 MIMO:

Multipleinput multipleoutput
 ML:

Maximum likelihood
 MMSE:

Minimal mean square error
 MPSK:

Mary phaseshift keying
 NMSE:

Normalized mean square error
 QPSK:

Quadrature phaseshift keying
 SNR:

Signaltonoise ratio
 TDD:

Time division duplexing
 ULA:

Uniform linear array
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Funding
This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 61601005, 61801114), Natural Science Foundation of Anhui Province (Grant Nos. 1808085MF164, 1608085QF138), Key Projects of the Outstanding Young Talents Program in the Universities of Anhui Province (Grant No. gxyqZD2016027), the Natural Science Foundation of Jiangsu Province (Grant No. BK20170688) and Doctoral Scientific Research Foundation of Anhui Normal University (Grant Nos. 2014bsqdjj38, 2018XJJ40).
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Mostly, I got the writing material from different journals as presented in the references. A MATLAB tool has been used to simulate my concept.
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XY conceived and designed the methods. XY performed the experiments and wrote the paper. AZ analyzed the simulation data. GZ and XG gave valuable suggestions on the structure of the paper. LY revised the original manuscript. All authors read and agreed the manuscript.
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Ye, X., Zhang, A., Zheng, G. et al. Multicell multiuser massive MIMO channel estimation and MPSK signal block detection applying twodimensional compressed sensing. J Wireless Com Network 2018, 238 (2018). https://doi.org/10.1186/s1363801812529
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Keywords
 Massive MIMO
 Sparse channel estimation
 Data block detection
 Twodimensional smoothed l _{0} (2DSL0)